16

I thought I was confident about what is meant by the slope of a line and how it relates to a rate of change, but I'm having doubts and I'm hoping that someone will be able to help me clear them up.

As I understand it, intuitively the slope of a line is a number $m$ that quantifies the amount it inclines from the horizontal, and its direction. Now, if $y$ is a function of $x$, such that $y=mx+c$, then one can quantify the rate of change in $y$ as one changes $x$ via the the ratio of their coordinate differences, such that $$m=\frac{\Delta y}{\Delta x}$$ Heuristically, can one (hopefully correctly) understand this quantity as follows:

The value of the function $y$ changes by an amount of $\Delta y$ units for every $\Delta x$ units change in $x$. Therefore, the value of $y$ changes by an amount $\frac{\Delta y}{\Delta x}$ units per unit change in x, which is exactly the rate of change in $y$ with respect to $x$, since it quantifies the amount the value of $y$ changes per unit change in $x$.

Would this be a correct understanding at all?

Jyrki Lahtonen
  • 140,891
Feyn_example
  • 311
  • 3
    Yes, it's perfectly correct. – Bernard Oct 19 '16 at 14:33
  • Very nicely written, along with being "spot on" (correct)! – amWhy Oct 19 '16 at 14:34
  • Agreed with AmWhy above, you've nailed it. Well written. –  Oct 19 '16 at 14:35
  • 2
    If only my classes consisted of students capable of writing this coherently about slopes. They often have a general idea, and perhaps remember a key phrase or two, but being able to put it into words this way is something else entirely. – Arthur Oct 19 '16 at 14:35
  • @Arthur Thank you for your kind comments, I really appreciate the encouragement, makes it much easier to motivate myself to continue my mathematical learning! I wanted to make sure that I was actually understanding what the maths is "saying" before moving on to more advanced concepts, rather than taking it for granted and running into troubles later on. – Feyn_example Oct 19 '16 at 14:44
  • Thanks for the kind comments and words of encouragement Benard, AmWhy and Bacon! (I couldn't figure out how to tag you all in one comment, so I hope this gets to all of you) – Feyn_example Oct 19 '16 at 14:47
  • @amWhy Thanks for the kind words. Just to check, in particular, is the last paragraph, on the intuition for why the rate of change is given by $\frac{\Delta y}{\Delta x}$, is what I wrote a good enough understanding to have when moving on to studying more advanced topics such as calculus? – Feyn_example Oct 19 '16 at 14:50
  • @Feyn, absolutely yes! – amWhy Oct 19 '16 at 14:52
  • @amWhy Great. Thanks for your help! – Feyn_example Oct 19 '16 at 14:55
  • 3
    You're always welcome, @Feyn. FYI, it is very, very, very rare for a newcomer's first posted question to receive 6 upvotes (and counting...), so you have every reason to feel good about it! – amWhy Oct 19 '16 at 14:55
  • As others already said, you are quite right. Let me add a comment.

    An interesting property of the slope of a line is that it is a constant. Wherever you measure it and with however small or large increments, you always find the same value.

    This is not true for curves and you will later learn how to deal with their slope. The general idea is that if you look at a smooth curve with a microscope, you essentially see a straight line, and you can define a slope locally, which can differ at every point.

     –  Oct 19 '16 at 15:02
  • @YvesDaoust For a curve a guess you approximate it around a particular point by a straight line and then the closer you are to this point the better the approximation becomes, such that the slope of the straight line is approximately equal to the slope of the curve near the chosen point?! – Feyn_example Oct 19 '16 at 15:42
  • @Feyn_example: yep, microscope. –  Oct 19 '16 at 15:49
  • @amWhy I hope you don't mind me asking, but I was wondering if you would mind checking this alternative explanation I've thought of?! Here it is: Suppose that $y$ is a function of $x$, then a change of $\Delta x$ units in the variable $x$ will induce a change in $y$ by an amount $\Delta y$ units. If there is a linear relationship between $x$ and $y$, then these two changes will be proportional, i.e. $\Delta y\propto\Delta x$ (if the relationship wasn't linear, then this wouldn't be true since the relationship between the changes would change depending on the size of $\Delta x$)... – Feyn_example Oct 19 '16 at 18:34
  • ... consequently, $\Delta y =m\Delta x$, where $m$ is the constant of proportionality between the two. Intuitively, this is saying that for every $\Delta x$ units change in $x$, $y$ changes by an amount $m\Delta x$ units, which implies that $y$ changes at a rate of $\frac{\Delta y}{\Delta x}$ units per unit change in $x$. – Feyn_example Oct 19 '16 at 18:39
  • @amWhy I've also heard $\frac{\Delta y}{\Delta x}$ being referred to as the average rate of change of a function. In what sense is it an average? To me an average of a set of data points is a central value around which those points cluster, and is found by summing the data points and then dividing this sum by the total number of data points. However, I don't see how the two concepts tally up?! ... – Feyn_example Oct 19 '16 at 21:22
  • ... Is it simply because, in the former one, is dividing the total change in $y$ into $\frac{\Delta y}{\Delta x}$ units of equal length, similar to how in the latter, the average value is found by dividing up the total value of the sum into $n$ intervals of equal length (where $n$ is the total number of data points) – Feyn_example Oct 19 '16 at 21:23
  • @Feyn Yes indeed (to your last comment immediately above). The smaller the intervals of change in x, the more accurate the "slope" is at a point in the interval. Indeed, you will find that through the use of the derivative of a function, one can determine an instantaneous value of the "rate of change" of $y$ with respect to $x$ of a function y= f(x): denoted $\frac{dy}{dx}$. Let your mind rest. You've done good work, and the questions you've asked and the curiosity you've shown are all the more reason to assure you that you are very well prepared to take the next step in your studies. – amWhy Oct 19 '16 at 21:35
  • @amWhy Ok great. Thanks for your patience and help! – Feyn_example Oct 20 '16 at 07:14
  • @Arthur - I have a question pertaining to this post: suppose that the slope is $m = \frac{2}{3}$, then is saying “the rate of change is two-thirds units of $y$ per $1$-unit of $x$” equivalent to saying “the rate of change is two units of $y$ per three units of $x$“? I believe these statements are equivalent since they both express how $y$ is changing with respect to $x$, but I just wanted to check. – Taylor Rendon Jan 20 '21 at 15:22
  • 1
    @TaylorRendon That sounds sound to me. The proportionality between the $x$ and $y$ movement is what matters, not their actual sizes. When talking about slope as a single number, we standardize to an $x$ movement of $1$ (or more correctly, we state the constant of proportionality). In particular, it is a good idea to phrase it (as you have done) as the motion in $x$ being the driving mechanism and the motion in $y$ being the result. – Arthur Jan 20 '21 at 15:26
  • @Arthur - Gotcha. Just to make sure I understand, when you say ‘that sounds sound to me’, you mean ‘that sounds correct/good to me’? (I’m not being pretentious, I just have not used that word as an adjective before now.) (-: – Taylor Rendon Jan 20 '21 at 19:49
  • 1
    @TaylorRendon Yes, "sound" is, among other things, an adjective that roughly means "correct". There is some nuance to it that I'm not entirely confident in, but that's the first of it. It just felt punny in that particular sentence. – Arthur Jan 20 '21 at 19:57

1 Answers1

5

Yes, your interpretation of slope is correct. In fact, some people take this interpretation as the definition of the slope of a line.

tc216
  • 903